What is the product of the polynomials (x2y+2)(x2−y) ?(1 point)

Question

Responses

x4y+x2−2y
x superscript 4 baseline y plus x squared minus 2 y

x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y

x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y

x3y−x2y2+2x2−2y

Answer 1

To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will apply the distributive property (also known as the FOIL method for binomials).

Let's distribute each term in the first polynomial to each term in the second polynomial:

1. Multiply \(x^2y\) by \(x^2\): \[ x^2y \cdot x^2 = x^4y \]

2. Multiply \(x^2y\) by \(-y\): \[ x^2y \cdot (-y) = -x^2y^2 \]

3. Multiply \(2\) by \(x^2\): \[ 2 \cdot x^2 = 2x^2 \]

4. Multiply \(2\) by \(-y\): \[ 2 \cdot (-y) = -2y \]

Now combine all these results: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

Thus, the final expression after combining all terms is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

So the product of the polynomials \((x^2y + 2)(x^2 - y)\) is: \[ \boxed{x^4y - x^2y^2 + 2x^2 - 2y} \]